[erikerikson]

In fact, I taught my child multiplication and negative numbers without running him through a lot examples. He publicly demonstrated the ability to multiply (by multiplying numbers during conversations) before his school ever taught it via drilling (in their case the Singapore method).

I agree about the shift in university. I really enjoyed what felt like a removal of the noise. Of course, the interconnection of the pieces and learning to instantiate the concepts into concrete are expected work of students. That all said, abstraction (or models) are lossy compressions and noticing their failures is essential to avoid becoming detached and/or delusional.